A Revival of Sylvester’s Wave Theory of Partitions
Let p(n) denote the number of partitions of the integer n. The first exact formula for p(n) was given by Hardy and Ramanujan in 1918. Their formula is a divergent series, which can be truncated in the appropriate place to give the exact value of p(n). Rademacher improved Hardy and Ramanujan's formula in 1938 to a rapidly converging infinite series. In early 2011, Ono and Bruinier announced a new formula which expresses p(n) as a finite sum of algebraic numbers.
The Hardy-Ramanujan-Rademacher formula is really a statement about the coefficients of a certain modular form whose coefficents happen to be the values of p(n). The Ono-Bruinier formula expresses p(n) as a sum of singular moduli of a certain weak Maass form that can be described in terms of the Dedekind eta function and the quasimodular Eisensten series E_2. Although p(n) is clearly a combinatorial function, neither of these formulas is combinatorial.
In this talk, I will attempt to show that J. J. Sylvester (1857) and J. W. L. Glaisher (1909) were well on their way to finding a combinatorial formula for p(n), and that a revival of their work combined with the power of modern computers could lead to a new formula for p(n).
DIMACS - Rutgers Summer Seminar Series in Combinatorics and Experimental Math
"A Revival of Sylvester’s Wave Theory of Partitions."
Mathematical Sciences Faculty Presentations.